Compositions and convex combinations of asymptotically regular firmly nonexpansive mappings are also asymptotically regular

Wang, Bauschke, Martín-Márquez Victoria, Moffat, Wang Xianfu

Description

Because of Minty's classical correspondence between firmly nonexpansive mappings and maximally monotone operators, the notion of a firmly nonexpansive mapping has proven to be of basic importance in fixed point theory, monotone operator theory, and convex optimization. In this note, we show that if finitely many firmly nonexpansive mappings defined on a real Hilbert space are given and each of these mappings is asymptotically regular, which is equivalent to saying that they have or "almost have" fixed points, then the same is true for their composition. This significantly generalizes the result by Bauschke from 2003 for the case of projectors (nearest point mappings). The proof resides in a Hilbert product space and it relies upon the Brezis-Haraux range approximation result. By working in a suitably scaled Hilbert product space, we also establish the asymptotic regularity of convex combinations. 2010 Mathematics Subject Classification: Primary 47H05, 47H09; Secondary 47H10, 90C25.

Sujets

• Hilbert space
• Metric map
• Resolvent

Informations

• Publié par: biomed
• Publié le: 01 janvier 2012
• Nombre de lectures: 19
• Langue: English

Extrait

Bauschkeet al.Fixed Point Theory and Applications2012,2012:53 http://www.fixedpointtheoryandapplications.com/content/2012/1/53

R E S E A R C HOpen Access Compositions and convex combinations of asymptotically regular firmly nonexpansive mappings are also asymptotically regular 1* 21 1 Heinz HBauschke ,Victoria MartínMárquez , Sarah MMoffat andXianfu Wang

* Correspondence: heinz. [e-mail] 1 Mathematics, University of British Columbia, Kelowna, BC V1V 1V7, Canada Full list of author information is available at the end of the article

Abstract Because of Minty’s classical correspondence between firmly nonexpansive mappings and maximally monotone operators, the notion of a firmly nonexpansive mapping has proven to be of basic importance in fixed point theory, monotone operator theory, and convex optimization. In this note, we show that if finitely many firmly nonexpansive mappings defined on a real Hilbert space are given and each of these mappings is asymptotically regular, which is equivalent to saying that they have or “almost have”fixed points, then the same is true for their composition. This significantly generalizes the result by Bauschke from 2003 for the case of projectors (nearest point mappings). The proof resides in a Hilbert product space and it relies upon the BrezisHaraux range approximation result. By working in a suitably scaled Hilbert product space, we also establish the asymptotic regularity of convex combinations. 2010 Mathematics Subject Classification:Primary 47H05, 47H09; Secondary 47H10, 90C25. Keywords:asymptotic regularity, firmly nonexpansive mapping, Hilbert space, maxi mally monotone operator, nonexpansive mapping, resolvent, strongly nonexpansive mapping

1 Introduction and standing assumptions Throughout this article, Xis a real Hilbert sace with innerroduct ∙ ,∙(1) and induced norm ||⋅||. We assume that m∈2, 3, 4,. . .andI:= 1,2,. . .,m (2) Recall that an operatorT: X®Xisfirmly nonexpansive(see, e.g., [13] for further 2 information) if (∀xÎX)(∀yÎX) ||TxTy||≤〈xy, TxTy〉and that a setvalued operatorA: X⇉Xismaximally monotoneif it ismonotone, i.e., for all (x, x*) and (y, y*) in the graph ofA, we have〈x  y, x*  y*〉≥0 and if the graph ofAcannot be prop erly enlarged without destroying monotonicity (We shall write domA= {xÎX | Ax≠ Ø} for thedomainofA, ranA=A(X) =∪ xÎX Axfor therangeofA, and grAfor the graphofA.) These notions are equivalent (see [4,5]) in the sense that ifAis maximally 1 monotone, then itsresolvent J A : = (Id +Afirmly nonexpansive, and if) isTis firmly © 2012 Bauschke et al; licensee Springer. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.